An algebraic closure of a field F is a field K such that: 1) K/F is an algebraic field extension,...

Question

Question

An algebraic closure of a field F is a field K such that: 1) K/F is an algebraic field extension,...

An algebraic closure of a field F is a field K such that: 1) K/F is an algebraic field extension, and 2) every nonconstant polynomial in K[x] has a root in K. If K is an algebraic closure of F, prove that every polynomial p(x) ∈ F[x] splits in K[x].

math Advanced-Math

Add a comment Improve this question Transcribed image text

Answer 1

Answer #1

Date p has a noot in K

Add a comment

Answer 2

Similar Homework Help Questions

Let k C F = kla) be a simple algebraic extension. Prove that F is normal...

Let k C F = kla) be a simple algebraic extension. Prove that F is normal over k if and only if for every algebraic extension FC K and every o E Autk(K),(F) = F.
Example of a normal extension which is not an algebraic closure of F

Example of a normal extension which is not an algebraic closure of F
Let E = F(a) be a (simple) extension of F. wherea E E is algebraic over F. Suppose the degree of ...

Let E = F(a) be a (simple) extension of F. wherea E E is algebraic over F. Suppose the degree of α over F is n Then every β E E can be expressed uniquely in the form β-bo-b10 + +b-1a-1 for some bi F. (a) Show every element can be written as f (a) for some polynomial f(x) E F (b) Let m(x) be the minimal polynomial of α over F. Write m(x) r" +an-11n-1+--+ n_1α α0. Use this...

9. Let E be an extension field of a field F. (1) What does it mean...

9. Let E be an extension field of a field F. (1) What does it mean for an element z EE being algebraic over F? (2) What does it mean for an element z E E being transcendental over F? (3) Can you find any element r e C such that r is transcendental over Q? (4) Can you prove that if a E E is algebraic over F then (F(a): F] is finite? (5) Can you prove that if...
Let k be a field of positive characteristic p, and let f(x)be an irreducible polynomial. Prove...

Let k be a field of positive characteristic p, and let f(x)be an irreducible polynomial. Prove that there exist an integer d and a separable irreducible polynomial fsep (2) such that f(0) = fsep (2P). The number p is called the inseparable degree of f(c). If f(1) is the minimal polynomial of an algebraic element a, the inseparable degree of a is defined to be the inseparable degree of f(1). Prove that a is inseparable if and only if its...
Consider the polynomial f(x) = x p − x + 1 ∈ Zp[x]. (a) Let a be a root of f in some extension. Prove that a /∈ Zp and a + b is a root of f for all b ∈ Zp. (b) Prove that f is irreducible over Zp. [Hi...

Consider the polynomial f(x) = x p − x + 1 ∈ Zp[x]. (a) Let a be a root of f in some extension. Prove that a /∈ Zp and a + b is a root of f for all b ∈ Zp. (b) Prove that f is irreducible over Zp. [Hint: Assume it is reducible. If one of the factors has degree m, look at the coefficient of x m−1 and get a contradiction.]

plz solve this question. For an extension L/K, let a € L be an algebraic element...

plz solve this question. For an extension L/K, let a € L be an algebraic element over K. Find the minimal polynomial of 1/a over K with respect to the minimal polynomial ma.kx) of a.
A finite field is any finite extension of Fp := Z/pZ. The characteristic of a field F is the gene...

A finite field is any finite extension of Fp := Z/pZ. The characteristic of a field F is the generator of the kernel of the map ι : Z → F, ι(1) = 1. (a) Prove that there exist finite fields of order pnfor any prime p. We denote such a field Fpn. (b) Prove that Fpn has characteristic p. (c) Prove that the Frobenius map φ(a) = ap is an automorphism of Fpn . (d) If f(x) ∈ Fpn...
2u-5 8. Let w be a root of f(x) = r +2r - 6 over the...

2u-5 8. Let w be a root of f(x) = r +2r - 6 over the field Q. Consider z E Q(w). Find a, b, c, d e Q us + w-2 such that : a + bu + cu2 + du 9. Let E be an extension field of a field F. (1) What does it mean for an element z E E being algebraic over F? (2) What does it mean for an element z EF being transcendental...

(a) (b) True or False: If K is a field and u, v are algebraic elements of degrees m, n respectively in some extension field of K, then (K(u,v Explain. If you are not sure, explain what you are...

(a) (b) True or False: If K is a field and u, v are algebraic elements of degrees m, n respectively in some extension field of K, then (K(u,v Explain. If you are not sure, explain what you are not sure about. Let E be an extension field of K, and let S (u_1,..u_n) be a set of elements of E. Explain what it means for S to be a *basis* for E as a vector space over K. *...

An algebraic closure of a field F is a field K such that: 1) K/F is an algebraic field extension,...

Homework Answers

Add Answer to:
An algebraic closure of a field F is a field K such that: 1) K/F is an algebraic field extension,...

Post as a guest

Earn Coins

Let k C F = kla) be a simple algebraic extension. Prove that F is normal...

Example of a normal extension which is not an algebraic closure of F

Let E = F(a) be a (simple) extension of F. wherea E E is algebraic over F. Suppose the degree of ...

9. Let E be an extension field of a field F. (1) What does it mean...

Let k be a field of positive characteristic p, and let f(x)be an irreducible polynomial. Prove...

Consider the polynomial f(x) = x p − x + 1 ∈ Zp[x]. (a) Let a be a root of f in some extension. Prove that a /∈ Zp and a + b is a root of f for all b ∈ Zp. (b) Prove that f is irreducible over Zp. [Hi...

plz solve this question. For an extension L/K, let a € L be an algebraic element...

A finite field is any finite extension of Fp := Z/pZ. The characteristic of a field F is the gene...

2u-5 8. Let w be a root of f(x) = r +2r - 6 over the...

(a) (b) True or False: If K is a field and u, v are algebraic elements of degrees m, n respectively in some extension field of K, then (K(u,v Explain. If you are not sure, explain what you are...

An algebraic closure of a field F is a field K such that: 1) K/F is an algebraic field extension,...

Homework Answers

Add Answer to: An algebraic closure of a field F is a field K such that: 1) K/F is an algebraic field extension,...

Post as a guest

Earn Coins

Add Answer to:
An algebraic closure of a field F is a field K such that: 1) K/F is an algebraic field extension,...